on the formal consistency of theory and experiment

On the Formal Consistency of Theory and Experiment, with

Applications to Problems in the Initial-Value Formulation of

the Partial-Differential Equations of Mathematical Physics

DRAFT

Erik Curiel

October 8, 2010

This paper is a corrected and clarified version of Curiel (2005), which itself is in the main a distillation of themuch lengthier, more detailed and more technical Curiel (2004). I make frequent reference to that paper throughout

this one, primarily for the statement, elaboration and proof of technical results omitted here.I would like to thank someone for help with this paper, but I cant.

1

Theory and Experiment

Abstract

The dispute over the viability of various theories of relativistic, dissipative fluids is an-alyzed. The focus of the dispute is identified as the question of determining what itmeans for a theory to be applicable to a given type of physical system under given con-ditions. The idea of a physical theorys regime of propriety is introduced, in an attemptto clarify the issue, along with the construction of a formal model trying to make theidea precise. This construction involves a novel generalization of the idea of a field onspacetime, as well as a novel method of approximating the solutions to partial-differentialequations on relativistic spacetimes in a way that tries to account for the peculiar needsof the interface between the exact structures of mathematical physics and the inexactdata of experimental physics in a relativisticall invariant way. It is argued, on the ba-sis of these constructions, that the idea of a regime of propriety plays a central role inattempts to understand the semantical relations between theoretical and experimentalknowledge of the physical world in general, and in particular in attempts to explainwhat it may mean to claim that a physical theory models or represents a kind of physicalsystem. This discussion necessitates an examination of the initial-value formulation ofthe partial-differential equations of mathematical physics, which suggests a natural setof conditionsby no means meant to be canonical or exhaustiveone may require amathematical structure, in conjunction with a set of physical postulates, satisfy in orderto count as a physical theory. Based on the novel approximating methods developed forsolving partial-differential equations on a relativistic spacetime by finite-difference meth-ods, a technical result concerning a peculiar form of theoretical under-determination isproved, along with a technical result purporting to demonstrate a necessary conditionfor the self-consistency of a physical theory.

Erik Curiel 2 October 8, 2010


Contents

1 Introduction 4

2 Relativistic Formulations of the Navier-Stokes Equations 10

2.1 The Three Forms of Partial-differential Equation . . . . . . . . . . . . . . . . . . . . 102.2 Parabolic Theories and Their Problems . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Hyperbolic Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 The Breakdown of Partial-Differential Equations as Models in Physics . . . . . . . . 12

3 The Kinematical Regime of a Physical Theory 15

3.1 Kinematics and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Constraints on the Measure of Spatiotemporal Intervals . . . . . . . . . . . . . . . . 193.3 Infimal Decoupages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 The Kinematical Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Physical Fields 30

4.1 Algebraic Operations on the Values of Quantities Treated by a Physical Theory . . . 314.2 Inexact Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Algebraic Operations on Inexact Scalars . . . . . . . . . . . . . . . . . . . . . . . . . 414.4 Inexact Scalar Fields and Their Derivations . . . . . . . . . . . . . . . . . . . . . . . 514.5 Inexact Tensorial Fields and Their Derivations . . . . . . . . . . . . . . . . . . . . . 584.6 Inexactly Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.7 Integrals and Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.8 Motleys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Physical Theories 73

5.1 Exact Theories with Regimes and Inexact, Mottled, Kinematically Constrained Theories 755.2 Idealization and Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3 An Inexact, Well Set Initial-Value Formulation . . . . . . . . . . . . . . . . . . . . . 925.4 A Physically Well Set Initial-Value Formulation . . . . . . . . . . . . . . . . . . . . . 975.5 Maxwell-Boltzmann Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.6 The Consistency of Theory and Experiment . . . . . . . . . . . . . . . . . . . . . . 105

6 The Soundness of Physical Theory 106

6.1 The Comparison of Predicted and Observed Values . . . . . . . . . . . . . . . . . . . 1066.2 Consistent Maxwell-Boltzmann Theories . . . . . . . . . . . . . . . . . . . . . . . . . 1106.3 The Dynamical Soundness of a Physical Theory . . . . . . . . . . . . . . . . . . . . . 1126.4 Theoretical Under-Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7 The Theory Is and Is Not the Equations 116



It is a capital mistake to theorize before one has data.Sir Arthur Conan DoyleA Scandal in Bohemia

In theory, theres no difference between theory and practice. In practice, there is.Yogi Berra

Theory without experiment is philosophy.

Allison Myers

1 Introduction

In this paper, I intend to investigate a series of questions on the complex interplay between the the-oretician and the experimentalist required for a mathematical theory to find application in modelingactual experiments and, in turn, for the results of those experiments to have bearing on the shapingand substantiation of a theory. On the one hand, we have the rigorous, exact and often beautifulmathematical structures of theoretical physics for the schematic representation of the possible statesand courses of dynamical evolution of physical systems.1 On the other hand, we have the intuitive,inexact and often profoundly insightful design and manipulation of experimental apparatus in thegathering of empirical data, in conjunction with the initial imposition of a classificatory structure onthe mass of otherwise disaggregated and undifferentiated raw data gathered. Somewhere in betweenthese extremes lie the mutual application to and qualification of each by the other.

It is one of the games of the experimentalist to decide what theory to play with, indeed, whatparts of what theory to play with, in planning experiments and designing instruments for themand modeling any particular experimental or observational arrangements, in light of, inter alia, theconditions under which the experiment will be performed or the observation made, the degree ofaccuracy expected or desired of the measurements, etc., and then to infer in some way or otherfrom the exact, rigorous structure of that theory, as provided by the theoretician, models of actualexperiments so that he may explicate the properties of types of physical systems, produce predictionsabout the behavior of those types of systems in particular cirumstances, and judge whether or notthese predictions, based on the schematic models contructed in the framework of the theory, conformto the inaccurately determined data he gathers from those experiments. It is one of the games

1I follow the discussion of Stein (1994) here in my intended use of the term schematic to describe the way

experiments are modeled in physics. That paper served as much of the inspiration for the questions I address in

this paper, as well as for many of the ways I attempt to address the questions. Besides to that paper, I owe explicit

debts of gratitude for inspiration to Geroch (2001), Stein (npub, 1972, 2004), with all of which, I hope, this paper has

affinities, in both method and conclusions.



of the theoretician to abduct exact, rigorous theories from the inaccurately determined, looselyorganized mass of data provided by the experimentalist, and then to articulate the rules of playfor those theories, by, inter alia, articulating the expected kinds and strengths of couplings thequantities of the theory manifest and the conditions under which they are manifested, leaving itto the experimentalist to design in light of this information probes of a sort appropriate to thesecouplings as manifested under the particular conditions of experiments. Jointly, the two try to find,in the physical world, common ground on which their games may be played. No matter what onethinks of the status of these sorts of decisions and articulations in sciencewhether one thinks theycan ultimately be explained and justified in the terms of a rational scientific methodology or whetherone thinks they are, in the end, immune to rational analysis and form the incorrigibly asystematicbed-rock of science, as it wereit behooves us, at the least, to get clearer on what is being decidedand articulated, and on how those decisions and articulations bear on each other, if, indeed, theydo at all .

I will not examine the actual play of any current or historical theoreticians and experimentalistsin their attempts to find common, mutually fruitful ground on which to engage each other. I leavethose issues, fascinating as they are, to other, more competent hands. Neither will I examine allthe different sorts of games in which they engage in their respective practices, rather treating onlythose played in one small part of their common playground, that having to do with the comparisonof predicted and observed values of a system as it dynamically evolves for the purposes of testingand substantiating a theory on the one hand, and refining experimental methods and design onthe other. [*** For this latter, cf. the suggestion by Lee and Yang of the experiments that showedviolation of party; differentiate these more explicitly from the construction of theoretical modelsthat only use well-founded theory to predict, with no thought of substantiation, such as planningthe moon-shots ***]. I do not deal explicitly with others, such as predictions that have nothingto do with comparison to observations (for instance, the use of Newtonian gravity in calculatingtrajectories during the Apollo projects flights to the Moon), or the calculation of fundamentalproperties of physical systems based on theoretical models (for instance, the use of the quantumtheory of solids to calculate the specific heat of a substance). [*** Distinguish comparison toobservation from use of observation in these examplesfor in the moon-shots they surely alsocompared the observed results of previous moon-shots to, among other things, refine their methods ofprediction and characterization for future ones ***]. The extension of the methods and argumentsof this paper to those and other practices strikes me as straightforward, but the proof is in thepudding, which I do not serve here, and have, indeed, not thought much about preparing, so I willsay nothing more about it.

I will examine in this paper only what one may think of as the logical structure of the relationsbetween the practice of the theoretician and that of the experimentalist, and, a fortiori, of thosebetween theory and experiment. I do not mean to claim that there is or ought to be a single suchstructure sub specie ternitatis, or indeed that there is any such structure common to differentbranches of physics, or indeed even one common to a single branch that remains stable and viableover arbitrary periods of time, in different stages of the scientific enterprise. I intend to investigate



only whether one can construct such a structure to represent some idealized form of these relations.I am not, in this paper, interested in how exactly the experimentalist and the theoretician may makein practice the transitions to and fro between, on the one hand, inaccurate and finitely determinedmeasurements, and, on the other, the mathematically rigorous initial-value formulation of a systemof partial-differential equations, whether their exact methods of doing so may be justified, etc. Iam rather concerned with the brute fact of its happening, whether there is indeed any way at allof constructing with some rigor and clarity a model of generic methods for doing so. Having sucha model in hand would show that there need be no gross logical or methodological inconsistency intheir joint practice (even if there is an inconsistency in the way physicists currently work, which Iwould not pretend to hazard a guess at). Indeed, it is difficult to see, on the face of it, how one maycomprehend these two to be engaged in the same enterprise in the first place, difficult, indeed, tosee even whether these two practices are in any sense consistent with each other, since it is not evenclear what such consistency may or may not consist of.2 While I seriously doubt that any formalanalysis of the relations between theory and practice I or anyone else may propose could answer thisquestion definitively with regard to a real physical theory and its experimental applications, the sortof analysis I attempt to outline here, if successful, would perhaps have the virtue of underlining thesorts of considerations one must take account of in judging the consistency of a real theory and itsapplication to the world. This may seem a Quixotic project, at best, on the face of it, but I think Ican say a few words in defense of its interest in the remainder of the introduction. In defense of itsfeasibility, I offer the paper itself.

Without a doubt, one can learn an extraordinary amount about a physical theory (and aboutthe world) by examining only its structure in isolation from the conditions required for its usein modeling phenomena, as is most often done in philosophical discussion of a technical natureabout physical theories in particular, and about the character of our understanding of the physicalworld in general. I will argue, however, mostly by example, that comprehensive understanding ofa physical theory will elude us unless we examine as well the procedures whereby it is employedin the laboratory, and, moreover, that comprehension of the nature of such knowledge as we mayhave of the physical world will similarly elude us without a serious attempt to understand both thetheoretical and the practical characters of that knowledge. In particular, the question I plan to

2Indeed, in my use of the terms experimentalist and theoretician throughout this paper, I am guilty of perpet-

uating the crudest of caricaturesas though the two lived in separate worlds, and had to travel some distance and

overcome great obstacles even to meet each other. Physicists such as Newton and Fermi, masters of both theory

and experiment, give my caricature the lie. Still, there is a grain of truth in the caricaturewhich is to say, that it

is a caricature, and so a fortiori it does strike home somewhat. When I was at the Relativity Group in the Fermi

Institute at the University of Chicago, the other graduate students and I used to say, only half jokingly, that other

groups of theoreticiansthose studying quantum field theory, or solid-state physics, for examplespoke a different

language than the one we spoke, and one had to work hard at translation to avoid a complete breakdown of com-

munication. One may extrapolate our feelings about experimentalists: if experimentalists could speak, we would

not understand them. (To which the experimentalist replied, When you speak, I cannot understand you!) [***

Even those physicists whose research focuses on one to the exclusion of the other yet share much in commonways

of thought, methods of argumentation, standards of proof, funds of knowledge, overarching goalswith those on the

other side of the aisle. ***].



address is not how one gets to a system of exact partial-differential equations from inaccurate data;nor is it how one gets from exact solutions of partial-differential equations to predictions that mayor may not accord with actually observed, inaccurate data (though this latter will be touched uponen passant to some degree). It is rather a question of the consistency of, perhaps the continuitybetween, the twoa question, if you like, of whether the theoretician and the experimentalist canbe understood as being engaged in the same enterprise,3 that of modeling and comprehending thephysical world, in complementary, indeed mutually supportive, ways. Another way of putting thepoint: philosophers, when having tried to understand the relation between theory and experiment,tend to have been vexed by the problem of how a theory gets into (and out of!) the laboratory, oftenframed in terms of the putatively inevitable theory-ladenness of observations; I am concerned herewith what one may call the converse problem, that of getting the laboratory into the theory, and thejoint problem, as it were, whether the theory and the laboratory admit at least in part a consistent,common model. Along the way, I will present an argument, in large part constituted by the bodyof the construction itself, that the initial-value formulation of the partial-differential equations ofa theory provides the most natural theater in which this sort of investigation can play itself out.Later in the paper, after the construction has been sketched, I will have more to say explicitly onthe privilege, as I see it, accruing to the role of the initial-value formulation in the comprehensionof physical theory.

I will focus the discussion around the idea of the regime of propriety4 of a physical theory(or physical regime or just regime, for short). From a purely extensive point of view, a regime of aphysical theory, roughly speaking, consists of the class of all physical systems cum environments thatthe theory is adequate and appropriate for the modeling of,5 along with a mathematical structureused to construct models of these systems, and a set of experimental techniques used for probingthe systems in a way amenable to modeling in the terms of that structure. It can be represented by,at a minimum

1. a set of variables representing physical quantities (the environment) not directly treated bythe theory but whose values in a given neighborhood are relevant to the issue of the theoryspropriety for use in modeling a particular physical system in that neighborhood, along with aset of algebraic and differential expressions formulated in terms of these variables, representingthe constraints these ambient, environmental quantities must satisfy in order for physicalsystems of the given type to be susceptible to treatment by the theory when they appear insuch environments

2. a set of algebraic and differential expressions formulated in terms of the variables and constantsappearing in the theorys system of partial-differential equations, representing the constraintsthe values of the quantities represented by those constants and variables must satisfy in or-der for the system bearing those quantities to be amenable to treatment by the theory; these

3I model use of the word enterprise in this paper on its use in Stein (2004).4I owe this term to Geroch (2001).5It is immaterial to my arguments whether one considers the class to comprise only existant systems or to comprise

as well possible systems, in whatever sense one wishes to give the modal term.



expressions may include as well terms from the set of variables representing relevant environ-mental quantities (kinematical constraints)

3. a set of algebraic expressions formulated in terms of variables representing the measure ofspatiotemporal intervals, constraining the character of the spatiotemporal regions requisite forwell-defined observations of the systems quantities to be performed in; these expressions mayinclude terms from the set of variables representing relevant environment quantities, as well asfrom the set of variables and constants appearing in the theorys system of partial-differentialequations (constraints on the systems characteristic spatiotemporal scales)

4. a set of methods for calculating the ranges of inaccuracy inevitably accruing to measurementsof the values of the systems quantities treated by the theory, depending on the sorts ofexperimental techniques used for probing the system, the environmental conditions underwhich the probing is performed, and the state of the system itself (including the stage ofdynamical evolution it manifests) at the time of the probingthese methods may include, e.g.,a set of algebraic and differential expressions formulated in terms of the variables and constantsappearing in the theorys system of partial-differential equations, the variables representing therelevant environmental factors, and the variables representing the measure of spatiotemporalintervals

5. a set of methods for calculating the ranges of admissible deviance of the predictions of thetheory on the one hand from actual measurements made of particular systems modeled bythe theory on the other, depending on the sorts of experimental techniques used for probingthe system, the environmental conditions under which the probing is performed, and the stateof the system itself (including the stage of dynamical evolution it manifests) at the time ofthe probingthese methods may include, e.g., a set of algebraic and differential conditionsformulated in terms of the variables and constants appearing in the theorys system of partial-differential equations, the variables representing the relevant environmental factors, and thevariables representing the measure of spatiotemporal intervals

The idea of a regime is perhaps best illustrated by way of an example. For the theory comprisingthe classical Navier-Stokes equations to model adequately a particular body of fluid, for instance,elements of its regime may include these conditions and posits:

1. the ambient electromagnetic field cannot be so strong as to ionize the fluid completely

2. the gradient of the fluids temperature cannot be too steep near equilibrium

3. only thermometric systems one centimeter in length or longer are to be used to measure thefluids temperature, and the reading will be taken only after having waited a few seconds forthe systems to have settled down to equilibrium

4. the chosen observational techniques to be applied, under the given environmental conditionsand in light of the current state of the fluid, yield data with a range of inaccuracy of 1%,with a degree of confidence of 95%



5. a deviance of less than 3% of the predicted from the observed dynamic evolution of the sys-tems temperature, taking into account the range of inaccuracy in measurement, is within theadmissible range of experimental error for the chosen experimental techniques under the givenenvironmental conditions, in light of the current state of the fluid

I neither promise nor threaten to offer in this paper a definitive analysis of the concept of a regimeor indeed of any of its constituents. I will rather sketch one possible way one may construct a(moderately) precise and rigorous model of the concept, with the aim of illuminating the sorts ofquestions one would have to answer in order to provide a more definitive analysis. The hope is thatsuch a model and correlative demonstration may serve as a contructive proof of the formal consistencyof the practice of the experimentalist and the practice of the theoretician in physics, indeed, as aconstruction of the common playground, as it were, of the two, playing with the toys and rides andgames of which we may pose precise questions of a technical nature about the interplay betweentheory and experiment, and attempt to answer such questions, at least in so far as one accepts theviability of the sort of formal model I construct.

The structure of the paper is as follows. In order to illustrate the sort of considerations thatmotivate and found my proposed definition and analysis of the idea of a regime, I begin, in 2,by briefly analyzing the dispute over hyperbolic reformulations of the theory of relativistic Navier-Stokes fluids, as the dispute illuminates the issues nicely. The points drawn from this analysis leadnaturally into the introduction in 3 of the notion of a regime, and the sketch of a construction ofa formal model purporting to represent the notion. In 4, I offer a mildly technical analysis of themathematical representation appropriate for the modeling of physical fields by theories with regimes,necessary for the culmination of the paper in 5, in which I analyze the initial-value formulation of thepartial-differential equations of theoretical physics (as opposed to that of those in pure mathematics)based on my analysis of the idea a regime, and draw several consequences from the analysis, andin 6, in which I discuss the criteria one may want to demand a theory satisfy in order for it to bethought empirically adequate. One of the most interesting of the results of this discussion describesa peculiar form of theoretical under-determination necessarily attendant on a physical theory, in sofar as the theory possesses a regime in the idealized sense proposed in this paper.

For the most part, I will deal only with the case of the interaction of theory and experimentwhen both the theoretical structures and the experimental practices are well worked out and wellunderstood; the investigation of those relations when one is dealing with novel theory, novel experi-ments, or both, presents far too many difficult and unavoidable questions for me to treat with anyadequacy or depth here.

The entire paper, if you will, may be considered an exercise in approximation and idealizationin the philosophy of physics in the attempt to work out part of its regime of propriety.



2 Relativistic Formulations of the Navier-Stokes Equations

2.1 The Three Forms of Partial-differential Equation

[*** Briefly characterize the elliptic, parabolic and hyperbolic forms of partial-differential equation,a` la Sommerfeld (1964). Define hyperbolization of an elliptic or parabolic system of equations.***]

2.2 Parabolic Theories and Their Problems

It is sometimes held that parabolic systems of partial-differential equations, such as the Navier-Stokessystem or Fouriers equation of thermal diffusion, do not have well set initial-value formulations.6

This, of course, depends on ones formulation of the idea of an initial-value formulation. Thefollowing is known, for example, about the Navier-Stokes system in non-relativistic physics (Temam1983, passim):

1. for appropriate initial data on a 3-space of absolute simultaneity, say t = 0, there exists a0 <


fluid is not zero.7 Kostadt and Liu (2000) have disputed the admissibility of this solution, claimingthat it arises from an ill-set initial-value formulation. They conclude that Landau and Lifschitzsparabolic formulation is in fact viable as a mathematical representation of a physical theory, at leastin so far as such objections go.

These discussions and arguments are exemplary of the problems faced by theoreticians whenattempting to model novel systems, or systems that can be investigated only with great difficulty.Of particular relevance for our study is the focus on whether or not the initial-value formulation ofthe partial-differential equations of a theory is well set or not. This notion will play an indispensablerole in the characterizations we offer, in 5, of a regime of propriety and a theory possessing one.

2.3 Hyperbolic Theories

While the problems mentioned in 2.2 have served as stimulus for finding a hyperbolic extension ofthe relativistic Navier-Stokes system, in the attempt it was suggested that there may be two otherperhaps even stronger reasons to find a viable such extension. In particular, it was suggested that,contrary to early assumptions, the hyperbolic theories might produce predictions differing from thoseof the parabolic system in certain tightly constrained circumstances in which both were applicable,offering the possibility of experimental tests of the hyperbolic systems.8 Even more enticingly, itwas suggested that the hyperbolic theories could be applied in circumstances in which the parabolicsystem becomes in one way or another inapplicable. I will briefly discuss how it is hoped thatthe hyperbolic systems may resolve the problems mentioned in 2.2, but my primary focus for themajority of the section will be on the two novel suggestions just mentioned.9

In order to discuss these issues further, it will be convenient to be more precise than there hasyet been call for. Fix a relativistic spacetime10 (M, gab). Then a relativistic Navier-Stokes fluid (orjust Navier-Stokes fluid, when there is no ambiguity) is a physical system such that:

1. its local state is completely characterized by the set of dynamical variables representing the7Their solution has its origin in the fact that Landau and Lifschitz (1975) define the mean fluid velocity by the

net momentum-fluxthe so-called kinematic velocityrather than by the flux of the particle-number densitythe

dynamic velocity. Whereas in classical physics these two quantities are identical, this is not generically the case in

relativity, though it may be in particular cases, such as for a system in complete thermodynamic equilibrium. See

Earman (1978) for a discussion.8Of course, given the profound observational entrenchment of the parabolic Navier-Stokes system, one of the

conditions demanded of such hyperbolic extensions will be that they (more or less exactly) recapitulate the predictions

of the original system under appropriate conditions.9For arguments in support of both suggestions, see Muller and Ruggeri (1993a), Herrera and Martnez (1997),

Anile, Pavon, and Romano (1998), Herrera, Prisco, and Martnez (1998), Herrera and Pavon (2001a), Herrera and

Pavon (2001b) and Jou, Casas-Vazquez, and Lebon (2001), et al. For attempts actually to conduct such studies,

see, e.g., Muller and Ruggeri (1993b), Zimdahl, Pavon, and Maartens (1996), Herrera and Martnez (1998), Jou,

Casas-Vazquez, and Lebon (2001), Eu (2002) and Herrera, Prisco, Martn, Ospino, Santos, and Troconis (2004).10For the purposes of this paper, a spacetime is a paracompact, Hausdorff, connected, orientable, smooth differential

manifold endowed with a smooth Lorentz metric under which the manifold is also time-orientable. The imposition of

temporal orientability simplifies presentation of the material dealing with the dynamic evolution of systems. It could

be foregone by restricting all analysis to appropriate subsets of spacetime.



mass density , particle-number density , mean fluid velocity11 a, heat flow qa, and shear-stress tensor ab, jointly satisfying the four kinematic constraints

ab = (ab) (2.3.1)

mam = nqn = 0 (2.3.2)

m(m) = 0 (2.3.3)m((+ p)am + pgam + 2q(am) + am) = 0 (2.3.4)

2. in the same physical regime, there are equations of state (specified once and for all), expressed interms of the dynamical variables characterizing the state, defining the pressure p, temperature , thermal conductivity , shear-viscosity and bulk-viscosity

3. in the same physical regime, these quantities jointly satisfy the two equations of dynamicevolution

qa + [(ma + am)(m log ) + nna] = 0 (2.3.5)ab + (m(a + (am)|m|b) + ( /3)(gab + ab)nn = 0 (2.3.6)

Equation (2.3.3) represents the conservation of particle number (all possible quantum effects arebeing ignored), (2.3.4) the conservation of mass-energy, (2.3.5) the flow of heat, and (2.3.6) theeffects of viscosity and stress.

2.4 The Breakdown of Partial-Differential Equations as Models in Physics

Classically, every Navier-Stokes fluid has a characteristic length (or equivalently, characteristic in-terval of time), the hydrodynamic scale, below which the description provided by the terms of thetheory breaks down. Typically there is only one such length, of the order of magnitude of the meanfree-path of the fluids particles; at this length scale, the thermodynamic quantities appearing in theequations are no longer unambiguously defined. Different sorts of thermometers, e.g., with sensi-tivities below the hydrodynamic scale, will record markedly different temperatures depending oncharacteristics of the joint system that one can safely ignore at larger scalesthe transparency ofeach thermometric system to the fluids particles, for instance. The other quantities fail in similarways.12

11I leave it purposely ambiguous as to which definition of fluid velocity appears, the so-called kinematic or dynamic,

as nothing hinges on it here.12It may seem that this sort of constraint on the definition of physical quantities manifests itself only as one shrinks

the germane spatial and temporal scales, but this is not so. Imagine the difficulties involved in attempting to define

what one means by the temperature of a cloud of gas three billion light years across. How will one, for instance,

calibrate the various parts of the thermometric apparatus, each with the others? There are many possible ways one

could conceive of doing it, with no guarantee that they will all yield the same answer. What if one wants to compute

the total angular momentum of the cloud in a particular direction? Or even just to compare the values of the

spin in a particular direction of two distant Hydrogen atoms? In general relativity, in a generic, curved spacetime,

there is no natural notion of the same direction at different points, and so a fortiori no natural method to identify

the same particular direction at different points of the spacetime to use in taking such averages or making such

comparisons.



There is no a priori reason why the definitions of all the different quantities, both kinematicand dynamic, that appear in the Navier-Stokes systembulk viscosity, shear viscosity, thermalconductivity, temperature, pressure, heat flow, stress distribution, and all the othersshould failat the same characteristic scale, even though, in fact, those of all known examples do, not only forNavier-Stokes theory but for all physical theories we have. This seems, indeed, to be one of themarkers of a physical theory, the existence of a single characteristic scale of length (equivalently:time, energy) for its kinematic and dynamic quantities, beyond which the definitions of them allfail.13 Clearly, if there were different characteristic lengths at which the definitions of differentquantities in the system broke down, the system itself would fail at the greatest such length-scale.Any phenomena that are observed at scales greater than the largest length at which one of thethermodynamic quantities becomes ill-defined are said to belong to the hydrodynamic regime; anyobserved below that scale are part of the sub-hydrodynamic regime (or the regime of moleculareffects).14

As Geroch (2001) points out, the Navier-Stokes system can fail in another way, at a length-scalelogically unrelated to the hydrodynamic length-scale, one at which the equations themselves mayfail to hold even though all the systems associated quantities remain well-defined. In other words,there may be a characteristic length-scale at which the expressions on the left-hand sides of theequations (especially the last two) may differ from zero by an amount, e.g., of the same order as thatof the terms appearing in the equations, while the equations remain valid at scales greater than thatlength-scale. I will refer to such a length-scale as the transient scale, gesturing at the fact that it isreasonable to expect that any such failures would have their origins in the dissipative fluxes of thefluids quantities transiently settling down as the quantities themselves approach their equilibrated,hydrodynamic values. This idea, in fact, inspires the preferred interpretations for the novel termsintroduced in the hyperbolic theories. I will refer to the greatest length-scale at which the systemfor any reason is no longer validwhether because the quantities lose definition or because theequations no longer holdas the break-down scale, and I will refer correlatively to the regime belowthis scale as the break-down regime.

Geroch (2001) points out a possible complication in the notion of a characteristic length-scale atwhich the system of equations breaks down (for whatever reason). The system may fail in a way more

13One could perhaps try to argue along the following lines to try to explain this fact. The definitions of all quantities,

kinematic and dynamic, fail at the same characteristic scale reflects the fact that these quantities and the relations

among them encoded in theorys equations of motion are all higher-level manifestations of the same underlying

phenomena, whatever hidden structure, beyond the reach of our theory, lies at the foundation of the phenomena at

issue. In the case of Navier-Stokes fluids, the underlying phenomena are those evinced by the statistical dynamics

of the molecular constituents of the fluid. The fluid density reflects the spatial, numerical distribution of the fluids

constituent molecules; changes in the fluid density arise from local, relative changes in the numerical distribution.

Pressure reflects the distribution of the molecules velocities, and changes in pressure, including the fluxes manifested

as stress and strain, arise from from local, relative changes in that distribution. The distribution of kinetic energy

among the molecules and its local, relative change evince temperature and heat. And so on. The relations among these

higher-level quantities encoded in the Navier-Stokes equations reflect the relations governing the statistical mechanics

of the quantities thus associated with the underlying distribution of the fluids molecular constituents.14This is also sometimes called the Knudsen regime, after [*** get cite ***].



complicated than can be described by a single, simple spatial or temporal length, or spatiotemporalinterval. As an example, he points out that relativity itself imposes constraints on the experimentalapplications of the theoretical model: the model must fail at every combined temporal and spatialscale, t and s respectively, jointly satisfying

s2 . t (2.4.1)

ands > ct (2.4.2)

where is the value of a typical dissipation coefficient for the fluid and c is the speed of light. Insteadof a characteristic break-down scale, this requirement defines a characteristic break-down area inthe t, s-plane. Note that the complement of this region of the plane, that is, the region in which thesystem remains valid (at least so far as these conditions are concerned), includes arbitrarily smalls-values and arbitrarily small t-values (though not both at the same time!).

In this terminology, proponents of hyperbolic theories contend that the examples they exhibitare of relativistic, dissipative fluids for which the parabolic system adequately models the equilib-rium behavior, yet which have transient scales measurably greater than their hydrodynamic scales,manifesting them in disequilibrated statesin other words, in certain kinds of disequilibrium, thequantities in the equations are well-defined, but the equations themselves fail to hold to a degreethat, for one reason or another, whether theoretical, experimental or pragmatic, is unacceptable.Geroch (2001), in turn, contends that there are no such fluids not even, as he puts it, any knowngedanken fluids.15 This is why such fluids represent an intriguing possibility: they would provide un-ambiguous examples of systems (presumably) amenable to theoretical treatment by the hyperbolictheories and (perhaps) accessible to experimental investigation.

Geroch (2001) offers an illuminating example of a particular way a system of equations may failwhile the quantities in terms of which the equations are formulated remain well defined. I call it theproblem of truncation, and, again, the hyperbolizations of the relativistic Navier-Stokes equationsprovide excellent illustrations. The hyperbolizations work, as we have said, by introducing terms ofsecond-order or higher,16 purportedly representing transient fluxes of the ordinary quantities treatedby the parabolic Navier-Stokes system. There is, however, no natural, a priori way to truncate theorder of terms one would have to include in the new equations to model the systems accuratelyenough, once one began including any higher-order terms, for the scales at which second-ordereffects become important, for instance, seem likely to be the same at which third-order, fourth-orderand 839th-order terms also may show themselves. It is, so far as I can see, a miracle and nothing morethat there are physical systems capable of being accurately modeled by the first-order Navier-Stokesequations, ignoring all higher-order effects.

As Geroch says,15He does exhibit an instructive but ultimately unsuccessful attempt to construct an example of such a Navier-Stokes

fluid.16The order of a term here refers roughly to the moment of the distribution function one must calculate to express

it.



The Navier-Stokes system, in other words, has a regime of applicabilitya limitingcircumstance in which the effects included within that system remain prominent whilethe effects not included become vanishingly small.

Geroch (2001, pp. 67)

The quantities modeled by the parabolic Navier-Stokes equations have a regime in which they aresimultaneously well-defined, satisfy the equations and have values stable with respect to higher-orderfluctuations. One cannot assume this for any amended equations one writes down, with novel termspurportedly representing higher-order effects. One must demonstrate it. On the face of it, this wouldbe a fools quest to attempt by a strictly theoretical analysis; in practice, it could be accomplishedonly through experimentation.

3 The Kinematical Regime of a Physical Theory

Philosophical analysis of particular physical theories, such as non-relativistic quantum mechanics,often focuses on the more or less rigorous mathematical consequences of the structure of the theoryitself, in abstraction from the necessary laboratory conditions required for application of the theoryin modeling the dynamic evolution of particular, actual systems. To clarify what I mean, considerthe usual schema of a Bell-type experiment considered by philosophers: an undifferentiated sourceof pairs of electrons in the singlet state, and an inarticulate, featureless Stern-Gerlach device tomeasure the spin of the electrons. This indeed constitutes a model of a physical system, but onlyin an abstract, even recherche`, sense. No consideration is given to the structure of the source ofthe electrons, the exact form of the coupling between the system under investigation (the pairs ofelectrons) and the instrument used to measure the relevant quantities of the system (the Stern-Gerlach device), or to the regime of propriety of the model they are using for this kind of systemscoupling with that sort of measuring apparatusit is a schematic representation of the experiment,in the most rarefied sense of the term.

It is taken for granted, for instance, that

1. the ambient temperature is not so high or so low as to disrupt the sources output of theelectrons

2. the electrons are not traveling so quickly (some appreciable fraction of the speed of light), norare the primary frequencies of the photons composing the magnetic field so high (having, e.g.,wave-lengths of the order of the Compton length), as to require the use of quantum field theoryrather than standard non-relativistic quantum mechanics in order to model the observationappropriately

3. the spins of the electron are measured using a Stern-Gerlach type of mechanism whose physicaldimensions are such as to allow its being treated as a classical device (as opposed to one whosedimensions are of such an ordera quantum-dot device, e.g.for which the measurement of



the electrons spin would become ambiguous, as one would have to account for the quantumproperties of the measuring device as well in modeling the interaction)

4. the metric curvature of the region in which the experiment is being performed ought not be sogreat as to introduce ambiguity in the assignment of correlations among the spin-componentsof different directions at the different spacetime points where the spin of each electron is,respectively, measured17

In the literature in general, no effort is put into determining how such restrictions may, if at all,affect the expected outcome of the experiments.18 While we are perhaps safe in blithely ignoringthese sorts of issues in the case of Bell-type experiments (and I am not even convinced of that),the study of theories of relativistic, dissipative fluids provides a clear example of a case in which wemay not safely ignore them, not only for reasons pertaining to the practice of physics but also, I feelsure, for reasons pertaining to the production of sound philosophical argument, as I intend to showin what follows.

The analysis of the debate over theories of a relativistic Navier-Stokes fluid shows that, at aminimum, the propriety of a theory for modeling a set of phenomena is constrained by conditionson the values of environmental quantities, the values of the quantities appearing in the theorysequations, and the measure of spatial and temporal intervals: a theory can be used to treat atype of physical system it putatively represents only when the systems environment permits thedetermination, within the fineness and ranges allowed by their nature, of the systems quantities overthe spatial and temporal scales appropriate for the representation of the envisioned phenomena. Inthis section, I will propose a possible model for dealing with these considerations precisely, thekinematical regime, requiring (in brief) with regard to the observation and measurement, and henceto the well-definedness, of the quantities treated by a theory:

1. a set of constraints on the measure of spatial and temporal intervals, and perhaps as well onthe behavior of the metric in general (e.g., that some scalar curvature remain bounded by agiven amount in the region)

2. a set of constraints on the values of the theorys quantities in conjunction with correlativeconstraints on environmental conditions

3. a set of methods for calculating the ranges of inevitable inaccuracy in the preparation or mea-surement of those quantities using particular sorts of experimental techniques under particularenvironmental conditions

17If the curvature were so great that parallel transport of a tangent vector along different paths from the point

of measurement of the spin of one of the electrons to the point of that of the other would yield markedly different

resultant tangent vectors, then the question of the correlation of the spins along opposite directions at the two

points becomes incoherent.18The analysis of Fine (1982) perhaps comes the closest in spirit in trying to take account of these sorts of issues with

regard to Bell-type experiments. The discussion of Stein (1972) presents a much richer, somewhat complementary

account to the one I sketch here.



I will not attempt to formulate any of the notions I discuss with rigor or to treat them to any depth.For those interested in the development of a rigorous technical apparatus for treating all theseissues, as well as for treating the issues raised in 4 and 5, see Curiel (2010b). The importanceof the regime, as I will argue later, is that much if not most of the semantic content of a theoryderives from the continual interplay between the theoretician and the experimentalist involved in itsaboriginal working out and the continual, ongoing work in its development as the theory matures andin its refinement as the theory settles into its maturation. That interplay constitutes the relevanceand importance of being able, at least in principle, for a theory to have the resources for the modelingof actual experiments, including the apparatus used in themit is only that process that renders toa theory one of the most important components of its semantic content. This is why I think it hasbeen to the detriment of philosophical comprehension of scientific theories that philosophers havenot focused more on the modeling of actual experiments, or, at least, have not focused more on thecollateral requirements involved in the construction of actual experimental models in the examplesthey tend to use in their philosphical argumentation.

3.1 Kinematics and Dynamics

[*** Give a brief characterization of both the kinematics and dynamics of a generic theory. ***]Before starting the analysis proper, we fix some definitions.19 Given a type of physical system, a

quantity of it is a (possibly variable) magnitude that can be thought of as belonging to the system,in so far as it can be measured (at least in principle) by an experimental apparatus designed tointeract with that type of system, in a fashion conforming to a particular coupling of the systemwith determinable features of its environment, which coupling may (at least, again, in principle)be modeled theoretically.20 Fix, then, a type of physical system, along with a system of partial-

19None of these prefatory definitions ought to be considered attempts at even the slovenly rigor, as it were, I aim

for in this paper, or, indeed, anything near it. These are rather in the way of marking off the field of play, much as

children determine a bit of a meadow as a soccer-field with episodic markers of the boundary (jackets, frisbees, . . . ),

which is to be interpolated between those markers as the niceness of the occasion demands.20This characterization of quantity involves (at least) one serious over-simplification. Not all quantities values can

be determined by direct preparation or measurement, even in principle, as this statement may suggest. Some, such

as that of entropy, can only be calculated from those of others that are themselves directly preparable or measurable.

Other quantities defy direct measurement for all intents and purposes, though perhaps not strictly in principle.

Consider, for example, the attempt to measure directly distances of the order of 1050 cmthe precision required ofany measuring device that would attempt it would demand that the probes it uses have de Broglie wave-lengths of

comparable scale, and so, correlatively, would demand the release of catastrophic amounts of energy in its interaction

with another systemthink of the energy of a photon whose wavelength was of that scale.

I am not sure whether the analysis I offer in this paper would or would not suffice for the treatment of these sorts

of quantity, though, offhand, I see no reason why it should make a difference. Temporal and spatial constraints do

not allow me to consider the question here, however.

Note that this is not an instrumentalist requirement. These measurements do not define the quantities, at least

not in all cases.

Perhaps this is not an inappropriate place to mention, EN passant, that, were one to allow oneself the momentary

luxury of Saturnalic speculation and wild extrapolation, it would be fun to imagine that the lack of such a thing, even

in principle, as an entropometer, and correlatively the lack of a unit of measure or scale for entropy, as the Joule



differential equations, which are intended to represent, among other things, the evolution of thesystems dynamic quantities over time. An interpretation of the equations in terms of the quantitiesof that type of physical system (or, more briefly: in terms of the physical system itself) is a complete,one-to-one correspondence between the set of variables and constants appearing in the equations onthe one hand and some sub-set of the known quantities of that type of physical system on the other,in conjunction with a set of statements describing the coupling of those quantities to known anddeterminable features of the environment precise and detailed enough to direct the experimentalist inconstructing probes and intruments tailored to the character of each quantity, as associated with thatkind of system, for its observation and measurement. The system of partial-differential equationsmodels the type of physical system if, given an interpretation of the equations in terms of that typeof physical system, and given any appropriate set of initial data for the equations representing apossible state of a physical system of that type, the mathematically evolving solution of the equationcontinues to represent a possible state of that system if it were to have dynamically evolved froma state represented by the initial data of the equations. In other words, the equations model thesystem if the equations solutions do not violate any of the systems inherent kinematic constraints.If, for example, a set of partial-differential equations as interpreted by the terms of a given typeof physical system predicted that systems of that type, starting from otherwise acceptable initialdata, would evolve to have negative mass, or would evolve in such a way that the systems worldlinewould change from being a timelike to being a spacelike curve, then we would likely conclude thatthose partial-differential equations do not model that type of system, at least not for that set ofinitial data. Note, in particular, that modeling is a strictly kinematical notion. The accuracy ofpredictions produced by the partial-differential equationswhether or not its solutions, under thegiven interpretation match to an admissible degree of accuracy the actual, dynamic evolution of suchsystemshas no bearing on the question of modeling at this stage. Let us say, then, that a physicaltheory comprises a system of partial-differential equations if those equations model the types ofsystems treated by the theory, under the interpretation the theory provides. For example, thetheory of relativistic Navier-Stokes fluids comprises equations (2.3.3)(2.3.6), under their standardinterpretation. Finally, by physical theory, I intend, very roughly speaking, an ordered set consistingof, at least,

1. a mathematical structure representing the states and the dynamical evolution of the physicalsystems treated by the theory (e.g., a space of states and a family of vector fields on it, thelatter representing the kinematically allowed evolutions of the system)

2. a set of experimental techniques for probing those systems

3. a mapping between the terms of the mathematical structure and the quantities associated withthose systems as observed and probed by the experimental techniques (an interpretation of

is to energy, somehow has to do with the fact that the Wigner time-reversal operator in quantum mechanics is not

a Hermitian operator but rather is anti-Hermitian. (The issue of the possible existence of an entropometer is, thank

goodness, not directly related to the possibility of a Maxwell demon, for the demon does not putatively measure the

entropy of a system, but rather only reduces it piece-meal.)



the mathematical structure in empiricial terms)

4. the set of data germane to knowledge of those quantities, collected from those systems bythe given experimental techniques and analyzed and informed by application of the givenmathematical structure

The fifth element I would include in the ordered set is a regime of propriety for the theory, to thearticulation of which I now turn.

I feel I need to make one last remark before proceeding, however. One may be tempted to thinkthat a fundamental physical theory, such as the quantum field theory of the Standard Model, oughtnot require specification of a regime for its applicability. This is not the case. Quantum field theorycan not solve in closed form the dynamical equations representing the evolution of arguably even thesimplest micro-system, the isolated Hydrogen atom. It rather relies on perturbative expansions, andthus requires the system to be not too far from equilibrium of one sort or another. Quantum fieldtheory in general, moreover, can not handle phenomena occurring in regions of spacetime in whichthe curvature is too large. The Standard Model breaks down in regimes far above the Planck scale.Not even quantum field theory formulated on curved-spacetime backgrounds can deal rigorously withphenomena under such conditions.21 Indeed, it appears that possession of a fairly well articulatedregime of propriety, as we will characterize it, or something nearly like it, is necessary for a theorysbeing viable as a theory of physics, as opposed to merely a chapter of pure mathematics.22 Bondi, ina paper on gravitational energy, puts his finger on the heart of the issue: Good physics is potentialengineering.23

3.2 Constraints on the Measure of Spatiotemporal Intervals

The idea of a regime, at bottom, rests on twin pillars: the idea that certain types of operationsassociated with the theory make sense (in some fashion or other) only when carried out over spa-tiotemporal regions whose dimensions satisfy certain constraints and in which some appropriatemeasure of the intensity of the metric curvature does not become too great; and that certain typesof operations associated with the theory make sense (in some fashion or other) only when the values

21I know of no theory of quantum gravity mature enough for it even to attempt the claim that it could do so.

Even if one could, and even were we able to observe and measure the peculiar quantities modeled by the theory, we

presumably would measure them using technological apparatus of some stripe, which, again presumably, would be

limited in its precision and its accuracy.

As an aside, I remark that I may appear to be leaving myself open to the charge of conflating two different ways in

which inaccuracy can accrue to measurments and predictions, one based on the nature of the quantities (as with the

statistical character of temperature, e.g., or as in the constraints imposed by the Heisenberg principle on quantum

phenomena), and the other based on de facto limitations due to the current stage of development of our technological

prowess. I should rather say that part of the point of this paper is to show that this distinction may not be so sharp

and clean as it appears at first glance. [*** discuss Newtons Third Rule of Reasoning from Principia, though not in

this footnote. ***].22The mathematician will balk at this merely, but she is not my primarily intended audience. Still, it would please

me were she able to read this paper with some profit, so I hope such rhetorical flourishes do not put her off too much.23Bondi (1962, p. 132). Italics are Bondis.



of some set of quantities relevant to systems treated by the theory satisfy certain constraints. Webegin with a few considerations about how one may constrain the measure of spatiotemporal inter-vals, which will culminate in a few quasi-technical definitions and results needed for the quasi-formalanalysis of the idea of a regime I intend to give.

Real initial data for real physical problems are not specified with arbitrary accuracy over anarbitrary region of a spacelike hypersurface in a relativistic spacetime. It is less of an idealizationto model initial-data as occupying a compact, connected region of spacetime, of non-zero metricalvolume, determined by the spatial extent of the system in conjunction with the temporal intervalduring which the measurement or preparation of the initial-data takes place. As we have seen inthe discussion of the dispute over the hyperbolic extensions to the Navier-Stokes system, moreover,the determinations of the values of real physical quantities appropriate for use in initial-data for agiven system will always be coarse-grained in the sense that they themselves are in some sense moreproperly modeled as pertaining to compact, connected sets of non-zero metrical volume, satisfyingcertain collateral metrical conditions, contained in the region occupied by the physical system, ratherthan as pertaining to individal spacetime points contained in the region occupied by the physicalsystem. Those sets, moreover, should be as small as possible in order to maximize the accuracy ofthe modeling of the experimental apparatus, while still being large enough to satisfy the constraintsthe theory places on the definition and measurement of its quantities and on the satisfaction of itsequations using the chosen methods of observation under the specified environmental conditions.

To study some physical phenomena modeled by a particular theory, then, we first need a compact,connected region of spacetime of non-zero metrical volume, which for the purposes of this discussionwe may without a great loss of generality assume to have properties as nice as we choose (wemay demand, e.g., that it be the closure of an open, convex, normal set), as the stage on whichthe phenomena will unfold and the experiment be played out. The theory may impose furtherrequirements on the region; it may demand, e.g., that its spatial and temporal dimensions (asdetermined in a specified manner) satisfy a set of algebraic constraints, or that the curvature in theregion satisfy a set of differential and algebraic conditions. Once so much is settled, the difficultylies in partitioning our region into components appropriate for the fixation of the values of thequantities modeled by the theory. Again, those components need to satisfy whatever constraintsthe nature of the quantities demand. It makes no sense in general to attempt to determine thetemperature of a system, e.g., on scales smaller than the mean free-path and the mean free-time offlight of the systems dynamically relevant constituents. For a sample of nitrogen gas under normalconditions on the surface of the Earth, for example, this would include the relevant measurementsof the nitrogen molecules, not of their electrons and nucleons, as calculated in a frame co-movingwith the surface of the Earth and not in one spinning wildly and moving at half the speed of lightwith respect to it. We therefore require that the individual regions to which values of temperatureare to be ascribed be larger than, in an appropriate sense, those characterized by the theorys break-down scales. Similarly, if we are to try to model a sample of nitrogen gas using the Navier-Stokesequations, for instance, then we must ensure that the dynamical evolution of the system is such thatthe gradient of its temperature on those scales not be too great just off points of equilibrium (as it



settles down to equilibrium, e.g., during preparation).A Maxwell-Boltzmann sort of partitioning of phase-space, and eo ipso of the spatiotemporal

region occupied by the system itself, into scraps of roughly equivalent volume and shape offers themost obvious way forward at this point.24 I do not find this solution satisfactory, howeveror,rather, I find it satisfactory for the particular treatment of the statistical mechanics of a more or lessideal gas, but I do not find it satisfactory for the generic treatment of the modeling of constraintson the determination of the values of quantities for many other kinds of physical theory. Althoughthermodynamics cum statistical mechanics provides the easiest and most straightforward examplesof the kinds of constraints that interest us, I would argue that such constraints form an integral partof the nitty-gritty of every physical theory, no matter how seemingly fundamental, as I gesturedat above.

Let us try to sketch a construction of a different sort of partitioning of a spatiotemporal region.25

Fix a compact, connected subset C of spacetime, of non-zero metrical volume, representing thespatiotemporal region in which the physical phenomena we would model play out. We demandthat such regions satisfy a few basic, generic, topological and metrical conditions, mostly alongthe lines of guaranteeing that the region is not too small along either its spacelike or its timelikedimensions, that its boundary is well-behaved, and so on. We will call such a region a canvas. Moreprecisely, a canvas is a convex, normal, compact, connected, 4-dimensional, embedded submanifoldof M.26 We will use canvases to model the spatiotemporal regions physical systems occupy in whicha specified family of observations and measurements occurs, as well as to model the elements intowhich such regions will be carved for the purpose of serving as points of the system to whichvalues of its associated quantities may be meaningfully ascribed (as opposed to ascribing the valuesof the quantities to points of spacetime itself).

To give a flavor of the sorts of algebraic conditions one may demand of the elements of such apartition, we first require terms in which to express the conditions. There is an endless supply oftheoretical terms one could employ to do so. I offer here only a sampling, by way of example. I do notthink that these have a preferred status over others one could propose. I offer them because they seemto me to be reasonably clear, to be easy to visualize and to have straightforward, meaningful physicalcontent. Other sets of terms could well serve better the purposes of a particular investigation. Suchchoices are, I think, fundamentally of a pragmatic and sthetical character. Choose, then, anelement O of the proposed partition of C and a point q on the boundary of O, and consider thefamily of all spacelike geodesics whose intersection with O (the interior of O) consists of a connectedarc one of whose points of intersection with the boundary of O is q. Calculate the supremum of theabsolute values of the proper affine length of all these arcs. Finally take the infimum of all these

24Synge (1957) has worked out such a device in illuminating detail for the statistical-mechanical treatment of ideal,

relativistic gases.25Again, for the rigorous details, see Curiel (2010b).26The full definition (see Curiel 2010b) includes the proviso that C not be a null 3-space with respect to the

spacetime metric. The exclusion of null hypersurfaces ensures that certain integrations and operations on the boundary

are always well defined. Since light never travels, so far as we know, in a true vacuum in any real physical situation,

this is a negligible exclusion for the goal of modeling real, inaccurate data over finite spatiotemporal regions.



suprema for every point q on the boundary. This is the infimal spacelike diameter of O. The infimaltimelike diameter is calculated in the analogous way, using timelike rather than spacelike arcs. Wetake the infimum of the suprema, as the simple infimum of the lengths for a Lorentzian metric wouldin general be zero as the arcs may approximate as closely as one wishes to a null arc. Note thatthe spacelike or the timelike infimal diameter of a connected set with non-zero metrical volume willalways be greater than zero (so long as the metric is well behaved, which we henceforth assume). Itthus follows directly from the definition of a canvas that its infimal spacelike and timelike diametersare always both greater than zero. Also, any 4-dimensional set with non-zero infimal spacelike andtimelike diameters has non-zero volume with respect to the spacetimes volume elementwe dealonly with measurable sets in this paperas one can always fit a non-trivial open set inside it (e.g.,a small tubular neighborhood of a geodetic, spacelike arc whose length is within some > 0 of theinfimal spacelike diameter). Thus it also follows that a canvas has non-zero measure with respect tothe volume element of spacetime. We will use these sorts of properties of canvases, especially thoserelating to their infimal diameters, to articulate the first kind of constraints a regime imposes ona theory, those directly addressing characteristic spatial and temporal measures of spatiotemporalregions appropriate for the application of the theory. [*** briefly sketch one possible way to thinkof the physical content of these diametersthat the longest way across the region for a particleor rod crossing near the center of the region will never be smaller than this amount ***]

It is not so easy to articulate terms in which the second half of the possible constraints onthe character of spatiotemporal regions appropriate for the definition of physical quantities, thosepertaining to the general behavior of the metric in the region, may be formulated. For instance,one can impose constraints on the intensity of the curvature in a region in any of a number of ways,from, say, fixing an upper bound on the total integral of any scalar curvature-invariant over theregion to fixing an upper bound on the average of such a scalar or an upper bound on the value ofthat scalar at any given point in the region; one may as well, for example, fix an upper bound onthe integrated components of the Riemann tensor as measured with respect to a parallel-propagatedframe-field along any timelike geodetic arc contained in the region; and so on. There are more generalsorts of considerations one may bring into play as well, including the imposition of some kind ofcausality conditions (e.g., that the region contain no almost closed, timelike curves), an exclusion ofcertain kinds of singular structure (e.g., that the region contain no incomplete timelike geodesics),a restriction on the topology of the spacetime manifold (e.g., that its second Stiefel-Whitney classvanish, the necessary and sufficient condition for a spacetime manifold to admit a globally defined,unambiguous spinor-structuresee Geroch 1969 and Geroch 1970b), and other general, metricalconsiderations (e.g., that the spacetime be asymptotically flat). I will not attempt to characterizewith any formality these sorts of metrical constraints, restricting myself mostly to speaking only ofconstraints on spatial and temporal measures, primarily because I see no way of doing so for theformer in light of their amorphous nature, not because I think they are unimportant or not worthconsidering. On the contrary, I think it would be of enormous interest to construct a formalism forstudying those sorts of constraints. In any event, the reader should bear in mind that, from hereon,when I speak of constraints on spatial and temporal measure I do not mean to exclude the other



sort from consideration. [*** remarkfor reasons like those sketched in Curiel (1999)that it isfar more difficult to lend clear, unambiguous physical content to constraints on the behavior of thecurvature and metric ***]

A set of algebraic constraints on the measure of temporal and spatial intervals, then, is a formalsystem of equations and inequalities with some number (greater than zero!) of unknown terms, eachterm representing a characteristic temporal or spatial scale associated with the quantities modeledby the theory. For the sake of simplicity, we will assume that, for any set of algebraic constraints ontemporal and spatial measures associated with the regime of a theory, there are only two unknownsused in all the expressions in the set, which we will interpret respectively as the spacelike and timelikeinfimal diameter of any region that is a candidate for having the values of the theorys quantitieslegitimately determined on it. A canvas satisfies such a set if its two diameters jointly satisfy theelements of the set. In the case of a relativistic Navier-Stokes fluid, for instance, we know that,for any element of a partition of the region it occupies, the infimal spacelike diameter ought to bestrictly greater than c times the infimal timelike diameter (see equation (2.4.2)). We also know thatthe two infimal diameters ought to be, respectively, at least of the order of the mean length of thefree path and the mean time of free flight of the fluids molecules, as determined in a reasonableframe.27

3.3 Infimal Decoupages

I shall now sketch the proposed manner of generically partitioning a canvas into elements to whichwe may apply our algebraic constraints. Fix a set of algebraic constraints on spatial and temporalmeasures and a canvas C satisfying the chosen constraints in such a way that the canvas containsas proper subsets other canvases also satisfying the constraints. A scrap S of the canvas is itself acanvas such that

1. it is a proper subset of C

2. it satisfies the constraints

3. its interior is topologically R4

The decoupage of a canvas is the family of all its scraps. A rich family of mathematical structuresaccrues to the decoupage in a natural way. It has, for instance, a natural topology under which itis Hausdorff, connected, and compact, if C itself is so. This topology can be extended to a -ringon which a Lebesgue measure, and thus integration of scalar fields, can be defined, grounded on thenatural Lebesgue measure associated with the spacetime metrics volume element. (See Curiel 2010bfor details.) The decoupage thus characterized has the structure of an infinite-dimensional space.

27With a little more effort, one can state this last condition in a relativistically invariant way, by stating it in

terms of the measure of intervals along and separations between timelike geodesics contained in the canvas such that

two otherwise free particles instantiating two given timelike geodetic arcs contained in the canvas will, with a given

probability, collide with a certain number of other particles traversing timelike geodetic arcs contained in the canvas

closer than the given distanceget it? Probably not. Simplify this mess.



I found it convenient for technical reasons in Curiel (2010b) to construct by the use of equivalenceclasses a finite-dimensional space capturing in approximate form all the essential structure of thedecoupage, and then to use this in place of the full decoupage. The construction of that derived spaceraises several interesting questions about the nature of the sorts of approximations one deals within physics, which we will not be able to address in this paper. In any event, from hereon, the termdecoupage will refer to that finite, approximative space rather than to the full, infinite-dimensionalspace; nothing in the papers arguments turn on the fact.

We will attempt to capture the idea of spatiotemporal regions whose dimensions satisfy certainconstraints, the ones appropriate for taking as the elements of the partition of the region in whichthe phenomena occur, by using decoupages. There are, again, several ways one may go about it.I will sketch only one. The following consideration will be our primary guide. On the one hand,the details of the physical state of the system on regions smaller than the break-down scale are,if not irrelevant, then at least ex hypothesi not sensibly representable in the theory at issue or donot yield results consonant with the solutions of the equations, whereas, on the other, those regionssignificantly larger than the break-down scale are not so fine-grained as one can in principle makethem for the purpose of maximizing the accuracy of observation and measurement. Given a theorywith its attendant set of algebraic constraints on spatial and temporal measure, we require a wayof specifying a family of subsets of a region that are in some sense or other as small as possiblewhile still conforming to the theorys contraints. In general, neither the set of spacetime pointsconstituting the region itself nor the whole decoupage itself of the region will serve the purpose.

Fix, then, a canvas C M and a set m of algebraic constraints on spatial and temporal intervals.The infimal decoupage of C, Cinf , consists of all the scraps of C whose volumes are, in a certainprecise sense, as small as possible while still being consistent with m. An infimal scrap is a member ofan infimal decoupage. Alternative definitions of an infimal decoupage could minimize, for instance,the volumes of the boundaries of the scraps, or a weighted average of the lengths of all the spacelikeand timelike arcs contained in each scrap, or the average scalar curvature of each scrap, or somecombination of these, and so on. I choose the definition based on volume not because I thinkit is a priori superior to the alternatives, but rather because it is simple, intuitively clear, andsuggestive of the usual Maxwell-Boltzmann partition of phase space in statistical mechanics. One ofthe alternatives could well fit the purposes of some particular analysis or investigation more closely.

It makes no sense to talk about the temperature, e.g., in regions on a scale finer than thatcharacteristic of the break-down of the modeling of the given system, nor indeed to speak of possiblesolutions to the partial-differential equations of a theory on a finer scale, for the equations are nolonger satisfied to the desired degree of accuracy in that regime, if they are well posed at all. Thisis why, in a substantive sense relevant to our purposes, a real-valued function whose domain isan infimal decoupage more appropriately models the details of the physical state associated withthe fluids temperature, e.g., than does a scalar field on a subset of spacetime: it forces one tofocus attention on those details and only those details both relevant to the experimental problem athand and sensical with respect to the formal structures of the theory being applied. For a Navier-Stokes fluid contained in a spacetemporal region, for example, the break-down scale, as discussed



in 2.4, defines part of the set of constraints on the spatial and temporal intervals over which thefluids quantities are well defined and over which the solutions to the equations themselves modelthe fluids actual dynamical evolution to the desired degree of accuracy, and so fixes the infimaldecoupage over the scraps of which the quantities associated with the fluid should be consideredfields.

Still, this all may sound more than superficially similar to the standard Maxwell-Boltzmannsort of partitioning. It differs from that device in important ways, however. Primary among thedifferences are two. First, in this scheme the scraps overlap in a densely promiscuous fashion. Thus,even though one can speak of, e.g., the temperature only on finite scraps rather than as associatedwith individual spacetime points, one can still speak of the temperature on such scraps arbitrarilyclose to each other in a topological sense. This may seem a slight advantage at best, but, as isshown in Curiel (2010b), exactly this aspect of the machinery developed here allows one, in completecontradistinction to the ordinary Maxwell-Boltzmann partition of phase space, to bring to bear withcomplete rigor the full battery of mathematical structures one most often employs in attackingboth theoretical and practical problems in physics, including topology, measure theory, differentialtopology, differential geometry, the theory of distributions and the theory of partial-differentialequations on finite-dimensional manifolds. One can then use these structure to articulate and proveresults of some interest (e.g., theorem 6.4.1 below) illuminating the relations among ordinary scalarfields as employed in theoretical physics and fields defined on these decoupages that, I argue, moreappropriately model the data gathered during the course of and used for the modeling of actualexperiments. Second, and at least as important, I do not see any other way of attempting to definesuch a partition in a relativistically meaningful and useful way. The standard Maxwell-Boltzmanndevice fixes the partition of the observatory once and for all into a finite lattice of scraps. Thispartition may provide excellent service for one observer but be next to useless, or worse, for another.The idea of the infimal decoupage allows one to take account of all such partitions all at once, as itwere, in an invariant manner.

3.4 The Kinematical Regime

Recall that the first type of failure of a theorys applicability to a given system, which we discussedin 2.4, stemmed from the ceasing to be well defined for one reason or another of the quantitiesthe theory attributes to the system. The International Practical Temperature Scale of 1927, asrevised in 1948, 1955 and 1960, provides an excellent, concrete example of this phenomenon.28 Forexample, the thermodynamical temperature scale between the primary fixed point 0.01o Celsius(the triple point of water at one standard atmosphere) and the secondary fixed point 630.5o Celsius(the equilibrium point between liquid and solid antimony at one standard atmosphere) is defined byinterpolation, using the variation in resistance of a standard platinum wire according to the equation

28Cf., respectively, Burgess (1928), Stimson (1949), Hall (1955) and Stimson (1961).



of Callendar (1887):29

t = 100(

Rt R0R100 R0

)+

(t

100 1)(

t

100

)where

R0 is the resistance of platinum as measured with the thermometer immersed in an air-saturated ice-water mixture at equilibrium, at which point the ice-point temperature is unaf-fected (to an accuracy of 0.001o Celsius) by barometric pressure variations from 28.50 inchesto 31.00 inches of mercury, and the resistance of the wire is independent of the static waterpressure up to an immersion-depth of 6 inches at sea-level

R100 is the resistance of platinum as measured with the thermometer immersed in saturatedsteam at equilibrium under atmospheric pressure (as determined using a hypsometer), thoughcorrections must be carefully made in this determination, the steam-point temperature be-ing greatly affected by variations in barometric pressure (for which, standard tables may beconsulted)

Rt is the resistance of platinum at temperature t (the temperature being measured), i.e., Rt isitself the quantity being measured that allows the calculation therefrom of the environmentstemperature

is a characteristic constant of the particular type of thermometer employed, defined at theprimary fixed point 444.6o Celsius (the equilibrium point between liquid and solid sulphur atone standard atmosphere)

Below 0.01o Celsius and above 630.5o Celsius, the Callendar equation quickly diverges from thethermodynamic scale. From 0.01o Celsius down to the primary fixed point -182.97o Celsius (theequilibrium point between liquid oxygen and its vapor at one standard atmosphere), the temperatureis also based on the resistance of a standard platinum wire, the interpolation being defined by anemendation of Callendars equation (transforming it from one quadratic to one cubic in the unknowntemperature), known as van Dusens equation; above 630.5o Celsius up to the primary fixed point1063.0o Celsius (the equilibrium point between liquid and solid gold at one standard atmosphere), thetemperature is based on the electromotive force generated by a 90%-platinum/10%-rhodium versus100% platinum thermocouple, the interpolation being defined by the so-called parabolic equationof thermocoupling; above 1063.0o Celsius, the temperature is based on the measurement of thespectrum of radiation by an optical, narrow-band pyrometer, the interpolation being defined byPlancks radiation formula.30 In all these cases, moreover, it is clear that one cannot speak of thetemperatures being measured on a spatial scale more finely grained than that corresponding to thephysical dimensions of the thermometric device employed, or on a temporal one more finely grained

29See, e.g., Benedict (1969, 4.14.4, pp. 249). This reference is not the most up-to-date with regard to the

international agreement on defining the standard, practical methods for the determination of temperature, but I have

found no better reference for the nuts and bolts of thermometry.30See, e.g., Benedict (1969, pp. 27).



than that of the time it takes the state of the device to equilibrate when placed in proper thermalcontact with the system under study, under the influence of fluctuations in the temperature of thesystem itself and its environment, under the given conditions.

As this example illustrates, the constraints on the definability and measurability of a quantity ina given theory must be variously given with regard to the parameters of particular types of systemsunder certain kinds of conditions, not generically once and for all in an attempt to constrain thedefinability and measurability of that quantity simpliciter. It is in part this very variability in thespecification of a quantitys definitionthat it is possible to make in a variety of waysthat leadsus to think that we have cottoned on to a real quantity, and not one artifactual of this particularexperimental arrangement under those particular conditions.31 This example makes clear, moreover,that in modeling different ranges of values of a given quantity different theories must be used. If onetreats phenomena in which temperatures rise above 1063 Celsius, for instance, ones theory mustinclude, or have the capacity to call upon the resources of, at least that part of quantum field theoryrequired for a Planckian treatment of electromagnetic radiation.32 We will therefore assume, at aminimum, that the range of admissible values for any quantity modeled by a theory is bounded bothfrom below and from above. In technical terms, this means that the family of scalar fields admissiblefor representing the distribution of the values of a quantity for any spatiotemporally extended systemtreated by the theory is itself uniformly bounded from below and from above. In a similar vein, weassume, roughly speaking, that the first several derivatives (the exact number being idiosyncratic toeach theory) of all the scalar fields are uniformly bounded from above and belowthere is no sense,for example, in using scalar fields that oscillate wildly in regions smaller than the breakdown scalewhen trying to represent a quantity.33

To make these ideas precise, fix a physical theory comprising a system of partial-differentialequations.

Definition 3.4.1 A kinematical regime of propriety of a theory (or a kinematical regime, for short)is an ordered quintuplet K (e, E, k, mk, K), where

1. e is the set of variables and constants the partial-differential equations of the theory are for-mulated in terms of

2. E is a finite set of variables and constants none of which appear in e, and thus not in any ofthe theorys equations

31See Newtons Third Rule of Natural Philosophy, at the beginning of Book III of the Principia, for a remarkably

concise and incisive discussion of a few facets of this issue, with particular emphasis on the nomination of certain

properties of a system as being simple or fundamental with regard to a theory.32This remark suggests that my use of theory does not entirely harmonize with standard usages in

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